schoolarithm

Arithmetic Problems for Elementary and Middle School Students

Description

The 23 fraction problems were presented to 191 first-level middle school students (about 11 to 12 years old). A subset of 13 problems is included in Stefanutti and de Chiusole (2017).

The eight subtraction problems were presented to 294 elementary school students and are described in de Chiusole and Stefanutti (2013).

Usage

data(schoolarithm)

Format

fraction17: a person-by-problem indicator matrix representing the responses of 191 persons to 23 problems. The responses are classified as correct (0) or incorrect (1).

The 23 problems were:

p01 % ( + ) : = ?
p02 % ( + ) - 2 = ?
p03 % ( + ) ( - ) = ?
p04 % ( + ) - = ?
p05 % + = ?
p06 % + = ?
p07 % + = ?
p08 % + = ?
p09 % + = ?
p10 % + = ?
p11 % + = ?
p12 % ( + ) = ?
p13 % 2 - = ?
p14 % (4 + - ) = ?
p15 % + =
p16 % - =
p17 % + =
p18 % + =
p19 % + = =
p20 Least common multiple \(lcm(6, 8) = ?\)
p21 % = ?
p22 % = ?
p23 % : = ?

subtraction13 is a data frame consisting of the following components:

School: factor; school id.
Classroom: factor; class room id.
Gender: factor; participant gender.
Age: participant age.
R: a person-by-problem indicator matrix representing the responses of 294 persons to eight problems.

The eight problems were:

p1 \(73 - 58\)
p2 \(317 - 94\)
p3 \(784 - 693\)
p4 \(507 - 49\)
p5 \(253 - 178\)
p6 \(2245 - 418\)
p7 \(156 - 68\)
p8 \(3642 - 753\)

Source

The data were made available by Debora de Chiusole, Andrea Brancaccio, and Luca Stefanutti.

References

de Chiusole, D., & Stefanutti, L. (2013). Modeling skill dependence in probabilistic competence structures. Electronic Notes in Discrete Mathematics, 42, 41–48. doi:https://doi.org/10.1016/j.endm.2013.05.144

Stefanutti, L., & de Chiusole, D. (2017). On the assessment of learning in competence based knowledge space theory. Journal of Mathematical Psychology, 80, 22–32. doi:10.1016/j.jmp.2017.08.003

Examples

library("pks")

data(schoolarithm)

## Fraction problems used in Stefanutti and de Chiusole (2017)
R <- fraction17[, c(4:8, 10:11, 15:20)]
colnames(R) <- 1:13
N.R <- as.pattern(R, freq = TRUE)

## Conjunctive skill function in Table 1
sf <- read.table(header = TRUE, text = "
  item  a  b  c  d  e  f  g  h
     1  1  1  1  0  1  1  0  0
     2  1  0  0  0  0  0  1  1
     3  1  1  0  1  1  0  0  0
     4  1  1  0  0  1  1  1  1
     5  1  1  0  0  1  1  0  0
     6  1  1  1  0  1  0  1  1
     7  1  1  0  0  1  1  0  0
     8  1  1  0  0  1  0  1  1
     9  0  1  0  0  1  0  0  0
    10  0  1  0  0  0  0  0  0
    11  0  0  0  0  1  0  0  0
    12  1  1  0  0  1  0  1  1
    13  0  0  0  0  0  1  0  0
")
K <- delineate(sf)$K  # delineated knowledge structure
blim(K, N.R)


Basic local independence models (BLIMs)

Number of knowledge states: 27
Number of response patterns: 115
Number of respondents: 191

Method: Minimum discrepancy
Number of iterations: 1
Goodness of fit (2 log likelihood ratio):
    G2(139) = 826.69, p = 0

Minimum discrepancy distribution (mean = 1.69634)
 0  1  2  3  4  5  6 
60 35 36 34 17  7  2 

Mean number of errors (total = 1.69634)
careless error    lucky guess 
     1.0924962      0.6038394 

Error and guessing parameters
       beta      eta
1  0.035470 0.247292
2  0.199750 0.050746
3  0.000000 0.047059
4  0.320030 0.092397
5  0.047000 0.187019
6  0.056800 0.123800
7  0.060740 0.168317
8  0.086440 0.220812
9  0.013050 0.244224
10 0.299410 0.006645
11 0.231360 0.008791
12 0.057850 0.120558
13 0.215260 0.080882

## Subtraction problems used in de Chiusole and Stefanutti (2013)
N.R <- as.pattern(subtraction13$R, freq = TRUE)

# Skill function in Table 1
# (f) mastering tens and hundreds; (g) mastering thousands; (h1) one borrow;
# (h2) two borrows; (h3) three borrows; (i) mastering the proximity of
# borrows; (j) mastering the presence of the zero; (k) mental calculation
sf <- read.table(header = TRUE, text = "
  item  f  g h1 h2 h3  i  j  k
     1  0  0  1  0  0  0  0  0
     2  1  0  1  0  0  0  0  0
     3  1  0  1  0  0  1  0  0
     4  1  0  1  1  1  0  1  0
     4  0  0  0  0  0  0  0  1
     5  1  0  1  1  1  1  0  0
     6  1  1  1  1  0  0  0  0
     7  1  0  1  1  1  1  0  0
     8  1  1  1  1  1  0  0  0
")
K <- delineate(sf)$K
blim(K, N.R)


Basic local independence models (BLIMs)

Number of knowledge states: 18
Number of response patterns: 78
Number of respondents: 294

Method: Minimum discrepancy
Number of iterations: 1
Goodness of fit (2 log likelihood ratio):
    G2(222) = 200.57, p = 0.84604

Minimum discrepancy distribution (mean = 0.64626)
  0   1   2   3 
149 103  39   3 

Mean number of errors (total = 0.64626)
careless error    lucky guess 
     0.4455788      0.2006808 

Error and guessing parameters
      beta      eta
1 0.073427 0.000001
2 0.124140 0.000001
3 0.076601 0.083135
4 0.000001 0.000001
5 0.103035 0.193084
6 0.045341 0.184319
7 0.042343 0.273775
8 0.048986 0.234973

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